Econ 20002_2022_SM1. Tutorial 1. Budget Constraints T1 sol. Page 1 of 5
ECON 20002. Intermediate Microeconomics
Tutorial 1 solutions
Consumer Theory: Budget Constraints

Problem 1. Luna Park math work-out is not fun
a) While a day pass gives you unlimited rides per day (you do not need to spend any extra
money after you got a pass), rides take time to undertake, and the available time is limited.
Thus, your choice set does not involve unlimited rides because you are constrained by time.
Choice sets involve any constraints on what you can or cannot do. Usually, we deal with
standard budget sets but more generally, choice sets are defined by constraints imposed by
money, time, laws, culture and any other constraints on our actions.
b) Altogether there are 71 ‘10-minute blocks’ of time available to you for rides (12 hours at six
blocks per hour, less the 10 minutes to get into the park. Thus, one end of the budget line is 71
rides on the Death; at the other end is 35.5 rides on the Ship.

The budget line is a line connecting these two corner points, for example, it includes a point
(31 rides on the Death, 20 rides on the Ship).
The slope of the budget line is given by the trade-off between Death and Ship rides: if you
sacrifice one Death ride (10 min), you can have 0.5 Ship rides (10 min / 20 min = 0.5), so the
slope is -0.5. Note that slope is negative.
Interior points inside the yellow triangle, e.g., 20 of each ride, are also in the choice set (time-
budget set). The time-budget set, therefore, is the whole yellow triangle.
The expression for the time-budget line is given by
10min * QDeath + 20min * QShip + 10min = 12h * 60min = 720 min.
Simplifying, we get QDeath + 2QShip = 71 .
Some students might note that it is difficult to have ‘half a ride’. A very good point! There is a discrete
version of the budget set with a bunch of dots. For example there are ‘dots’ at zero Death and 0, 1, 2,
… , 35 Ship, at 1 Death and 0, 1, 2, …, 35 Ship, at 2 Death and 0, 1, 2, …, 34 Ship, etc. Note that we
tend to use the continuous version (i.e. ignore integer issues) because it is easier but integer issues
should always be kept in mind and checked to make sure that answers ‘make sense’.

QShip
QDeath
35.5
71
time-budget line (time constraint) time-budget set
Econ 20002_2022_SM1. Tutorial 1. Budget Constraints T1 sol. Page 2 of 5
c) In this question, you now face two constraints. The time constraint given in part (a) still operates
but there is also a monetary constraint. You cannot spend all day on the Death even if you wanted
to. While you have enough time to go for 71 rides on the Death, you only have $120. Going for 71
rides on the Death will cost you $142, which is $22 more than your budget.
In fact, your budget will only allow you to have at most 60 rides on the Death (at $2 per ride). Or
you can choose 60 rides on the Ship, or 20 rides on the Death and 40 on the Ship, and so on.
The slope of the budget line is given by the trade-off between Death and Ship rides: if you give up
one Death ride ($2), you have just enough money to afford one Ship ride ($2); thus, the slope is
equal to -1.
The equation for the budget line is given by 2 * QDeath + 2 * QShip = 120.
Simplifying, we get QDeath + QShip = 60 .

If we draw both constraints (time and money) then we get the following graph.

The elements of the choice set must satisfy both constraints – you have to have enough time and
enough money (i.e., you need to be “on or under” both lines). Therefore, the choice set is given by
the shaded area below.

The two budget lines intersect at the point that satisfies both QDeath + 2QShip = 71 and QDeath + QShip = 60.
QDeath
QShip
35.5
60
71 60
budget line (monetary constraint); slope = -1
time-budget line; slope = - 0.5
Econ 20002_2022_SM1. Tutorial 1. Budget Constraints T1 sol. Page 3 of 5
Solving the two together: QShip = 60 − QDeath from the second equation;
substituting into the first one gives us QDeath + 2(60 − QDeath) = 71 => QDeath = 49.
Then QShip = 60 − QDeath = 60 − 49 = 11.



The budget kinked “line” is given by QDeath + 2QShip = 71, QDeath ≤ 49, QDeath + QShip = 60, QDeath > 49.


Problem 2. Electricity pricing

a) If Anita consumes less than or equal to 1,000 kWh of electricity per month, her budget
constraint is:
1 + 2 = => $0.10 + $1 = $400 => 0.1 + = 400 ≤ 1,000
Draw budget line as usual – she can afford 400/0.1=4,000 kWh electricity only or 400/1=400
other goods only; connect the corner points to get a budget line. The slope of the budget line is
(minus) price ratio: - 0.1/1 = - 0.1. Note that this line is applicable only for E ≤ 1,000!

Now assume that Anita consumes more than 1,000 kWh.
She pays $0.10 for the first 1000 kWh, but for anything above it (i.e. E - 1000), she will pay
only $0.05 per kWh.
Her budget constraint is: $0.10 ∗ 1000 + $0.05 ∗ ( − 1000) + $1 ∗ = $400 => 100 + 0.05( − 1000) + = 400 100 + 0.05 − 50 + = 400 0.05 + = 350 for > 1,000
Look at this – this budget line looks normal, but as if income = $350, price of electricity =
Econ 20002_2022_SM1. Tutorial 1. Budget Constraints T1 sol. Page 4 of 5
$0.05 per kWh, and the price of other goods = $1. Draw it as usual: Anita can afford 350/0.05
= 7,000 kWh electricity only or 350/1 = 350 other goods only; connect the corner points to get
a budget line. The slope of the budget line is (minus) price ratio: - 0.05/1 = - 0.05. Note that
this line is applicable only for E > 1,000!
You can check that when she consumes exactly E=1000, her consumption of other goods is
300 by using either of the two constraints. This is the point where two budget lines intersect.
First of them works only for E ≤ 1000 and the other one for E > 1000.
Put the two together to get budget constraint with a kink:





b) Electricity plan B gives us the same budget line as for Plan A, E ≤ 1000:
: 0.1 + = 400

Electricity plan C results in the following budget line:
1 + 2 = => $0.05 ∗ + $1 ∗ = $400 =>
: 0.05 + = 400

Econ 20002_2022_SM1. Tutorial 1. Budget Constraints T1 sol. Page 5 of 5
Draw budget line as usual – Anita can afford 400/0.05=8,000 kWh electricity only or
400/1=400 other goods only; connect the corner points to get a budget line. The slope of
the budget line is (minus) price ratio: - 0.05/1 = - 0.05. It is parallel to the budget line for
plan A when E > 1000, as prices are the same.



Electricity plan C is the best for Anita – she can afford all the bundles available under
plans A and B, and has many more to choose from (on her budget line).
Plan A: she can afford all bundles under plan B and some more bundles, so Anita would
prefer plan A to plan B. Note, though, that if she wants to consume less than (or equal to)
1,000 kWh of electricity, than both plans A and B give her the same options – so Plan A is
weakly better for her than plan B in that sense.